Nonlinear duality-invariant conformal extension of Maxwell’s equations
TL;DR: In this paper, all nonlinear extensions of the source-free Maxwell equations preserving both electromagnetic duality invariance and conformal invariance are found, and shown to be limits of a one-parameter generalization of Born-Infeld electrodynamics.
Abstract: All nonlinear extensions of the source-free Maxwell equations preserving both $SO(2)$ electromagnetic duality invariance and conformal invariance are found, and shown to be limits of a one-parameter generalization of Born-Infeld electrodynamics. The strong-field limit is the same as that found by Bialynicki-Birula from Born-Infeld theory but the weak-field limit is a new one-parameter extension of Maxwell electrodynamics, which is interacting but admits exact light-velocity plane-wave solutions of arbitrary polarization. Small-amplitude waves on a constant uniform electromagnetic background exhibit birefringence, but one polarization mode remains lightlike.
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TL;DR: In this article, it was shown that the resulting nonlinear theory preserves both conformal invariance and SO(2) duality-rotation invariance, and their result can be derived in a simpler way.
70 citations
Additional excerpts
...ngth Fµν is expressed in terms of vector potentials, Fµν = ∂µAν −∂νAµ, (2) and the dual of Fµν is defined by ∗Fµν = 1 2 ǫµνρσF ρσ. (3) Recently Bandos, Lechner, Sorokin, and Townsend have demonstrated [4] that such is indeed the case. This is a profound result. However, the line of reasoning in Ref. [4] may seem somewhat meandering. The aim of the present note is to come to the same upshot in a direct...
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TL;DR: In this paper , the deformation of the ModMax theory was investigated under the flow invariance for Born-Infeld theories, and it was shown that the deformed theory is the generalized non-linear Born-infeld electrodynamics.
61 citations
TL;DR: In this article, the relationship between nonlinear p-form electrodynamics with conformal-invariant weak-field and strong-field limits is clarified, with a focus on duality invariant (2n − 1)-form electdynamics and chiral 2n-forms in Minkowski spacetime.
Abstract: Relations between the various formulations of nonlinear p-form electrodynamics with conformal-invariant weak-field and strong-field limits are clarified, with a focus on duality invariant (2n − 1)-form electrodynamics and chiral 2n-form electrodynamics in Minkowski spacetime of dimension D = 4n and D = 4n + 2, respectively. We exhibit a new family of chiral 2-form electrodynamics in D = 6 for which these limits exhaust the possibilities for conformal invariance; the weak-field limit is related by dimensional reduction to the recently discovered ModMax generalisation of Maxwell’s equations. For n > 1 we show that the chiral ‘strong-field’ 2n-form electrodynamics is related by dimensional reduction to a new Sl(2; ℝ)-duality invariant theory of (2n − 1)-form electrodynamics.
46 citations
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TL;DR: In this paper, a supersymmetrization of any four-dimensional nonlinear electrodynamics theory with a Lagrangian density satisfying a convexity condition that relates to semi-classical unitarity is proposed.
Abstract: We give a prescription for $$ \mathcal{N} $$
= 1 supersymmetrization of any (four-dimensional) nonlinear electrodynamics theory with a Lagrangian density satisfying a convexity condition that we relate to semi-classical unitarity. We apply it to the one-parameter ModMax extension of Maxwell electrodynamics that preserves both electromagnetic duality and conformal invariance, and its Born-Infeld-like generalization, proving that duality invariance is preserved. We also establish superconformal invariance of the superModMax theory by showing that its coupling to supergravity is super-Weyl invariant. The higher-derivative photino-field interactions that appear in any supersymmetric nonlinear electrodynamics theory are removed by an invertible nonlinear superfield redefinition.
44 citations
TL;DR: In this paper, a self-gravitating solution to the ModMax theory was proposed, which is characterized by conformal symmetry and the S O (2 ) duality-rotation invariance.
35 citations
References
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01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
11,008 citations
TL;DR: In this paper, the authors compared Dirac's theory of the positron to those proposed by Born and showed that the field strength of large fields differs strongly from those of small fields.
Abstract: [arXiv:physics/0605038]: According to Dirac’s theory of the positron, an electromagnetic field tends to create pairs of particles which leads to a change of Maxwell’s equations in the vacuum. These changes are calculated in the special case that no real electrons or positrons are present and the field varies little over a Compton wavelength. The resulting effective Lagrangian of the field reads: $\cal{L} = \frac{\displaystyle 1}{\displaystyle 2} (\cal{E}^2 - \cal{B}^2) + \frac{\displaystyle e^2}{\displaystyle h c}\int_0^\infty e^{-\eta} \frac{\displaystyle d \eta}{\displaystyle\eta^3}\left\{ i \eta^2 (\cal{EB})\cdot \frac{\displaystyle\cos\left(\frac{\displaystyle\eta}{\displaystyle\vert\cal{E}_k\vert}\sqrt{\cal{E}^2 - \cal{B}^2 + 2i (\cal{EB})}\right) + conj.}{\displaystyle\cos\left(\frac{\displaystyle\eta}{\displaystyle\vert\cal{E}_k\vert}\sqrt{\cal{E}^2 - \cal{B}^2 + 2i (\cal{EB}})\right) - conj. } + \vert\cal{E}\vert^2 + \frac{\displaystyle\eta^2}{\displaystyle 3} (\cal{B}^2 - \cal{E}^2)\right\}$. $\cal{E}$, $\cal{B}$ field strengths. $\vert\cal{E}_k\vert = \frac{\displaystyle m^2 c^3}{\displaystyle e\hbar} = \frac{\displaystyle 1}{\displaystyle 137} \frac{\displaystyle e}{\displaystyle(e^2/m c^2)^2}$ critical field strengths. The expansion terms in small fields (compared to $\cal{E}$) describe light-light scattering. The simplest term is already known from perturbation theory. For large fields, the equations derived here differ strongly from Maxwell’s equations. Our equations will be compared to those proposed by Born. Original German abstract [Z.Phys. 98(1936)714]: Aus der Diracschen Theorie des Positrons folgt, da jedes elektromagnetische Feld zur Paarerzeugung neigt, eine Abanderung der Maxwellschen Gleichungen des Vakuums. Diese Abanderungen werden fur den speziellen Fall berechnet, in dem keine wirklichen Elektronen und Positronen vorhanden sind, und in dem sich das Feld auf Strecken der Compton-Wellenlange nur wenig andert. Es ergibt sich fur das Feld eine Lagrange-Funktion: $\cal{L} = \frac{\displaystyle 1}{\displaystyle 2} (\cal{E}^2 - \cal{B}^2) + \frac{\displaystyle e^2}{\displaystyle h c}\int_0^\infty e^{-\eta} \frac{\displaystyle d \eta}{\displaystyle\eta^3}\left\{ i \eta^2 (\cal{EB})\cdot \frac{\displaystyle\cos\left(\frac{\displaystyle\eta}{\displaystyle\vert\cal{E}_k\vert}\sqrt{\cal{E}^2 - \cal{B}^2 + 2i (\cal{EB}})\right) + konj}{\displaystyle\cos\left(\frac{\displaystyle\eta}{\displaystyle\vert\cal{E}_k\vert}\sqrt{\cal{E}^2 - \cal{B}^2 + 2i (\cal{EB})}\right) - konj } + \vert\cal{E}\vert^2 + \frac{\displaystyle\eta^2}{\displaystyle 3} (\cal{B}^2 - \cal{E}^2)\right\}$. ($\cal{E}$, $\cal{B}$ Kraft auf das Elektron. $\vert\cal{E}_k\vert = \frac{\displaystyle m^2 c^3}{\displaystyle e\hbar} = \frac{\displaystyle 1}{\displaystyle ,,137``} \frac{\displaystyle e}{\displaystyle (e^2/m c^2)^2}$ „Kritische Feldstarke“.) Ihre Entwicklungsglieder fur (gegen $\vert\cal{E}_k\vert$) kleine Felder beschreiben Prozesse der Streuung von Licht an Licht, deren einfachstes bereits aus einer Storungsrechnung bekannt ist. Fur grose Felder sind die hier abgeleiteten Feldgleichungen von den Maxwellschen sehr verschieden. Sie werden mit den von Born vorgeschlagenen verglichen.
2,059 citations
"Nonlinear duality-invariant conform..." refers background in this paper
...[3] W....
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...The Euler-Heisenberg equations [3], which incorporate vacuum polarisation effects of QED, is another....
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TL;DR: Aus der Diracschen Theorie des Positrons folgt, da jedes elektromagnetische Feld zur Paarerzeugung neigt, eine Abanderung der Maxwellschen Gleichungen des Vakuums as discussed by the authors.
Abstract: Aus der Diracschen Theorie des Positrons folgt, da jedes elektromagnetische Feld zur Paarerzeugung neigt, eine Abanderung der Maxwellschen Gleichungen des Vakuums. Diese Abanderungen werden fur den speziellen Fall berechnet, in dem keine wirklichen Elektronen und Positronen vorhanden sind, und in dem sich das Feld auf Strecken der Compton-Wellenlange nur wenig andert. Es ergibt sich fur das Feld eine Lagrange-Funktion:
$$\begin{gathered} \mathfrak{L} = \tfrac{1}{2}(\mathfrak{E}^2 - \mathfrak{B}^2 ) + \tfrac{{e^2 }}{{hc}}\int\limits_0^\infty {e^{ - \eta } } \tfrac{{d\eta }}{{^{\eta ^3 } }}\left\{ {i\eta ^2 (\mathfrak{E}\mathfrak{B}) \cdot } \right.\frac{{\cos \left( {\tfrac{\eta }{{\left| {\mathfrak{E}_k } \right|}}\sqrt {\mathfrak{E}^2 - \mathfrak{B}^2 + 2i(\mathfrak{E}\mathfrak{B})} } \right) + konj}}{{\cos \left( {\tfrac{\eta }{{\left| {\mathfrak{E}_k } \right|}}\sqrt {\mathfrak{E}^2 - \mathfrak{B}^2 + 2i(\mathfrak{E}\mathfrak{B})} } \right) - konj}} \hfill \\ \left. { + \left| {\mathfrak{E}_k } \right|^2 + \tfrac{{\eta ^2 }}{3}(\mathfrak{B}^2 - \mathfrak{E}^2 )} \right\}. \hfill \\ \left( {\begin{array}{*{20}c} {\mathfrak{E},\mathfrak{B} Kraft auf das Elektron.} \\ {\left| {\mathfrak{E}_k } \right| = \frac{{m^2 c^3 }}{{e\hbar }} = \frac{1}{{137}}\frac{e}{{({{e^2 } \mathord{\left/ {\vphantom {{e^2 } {m c^2 }}} \right. \kern-
ulldelimiterspace} {m c^2 }})^2 }} = Kritische Feldst\ddot arke.} \\ \end{array} } \right) \hfill \\ \end{gathered}$$
Ihre Entwicklungsglieder fur (gegen
$$\left| {\mathfrak{E}_k } \right|$$
kleine Felder beschreiben Prozesse der Streuung von Licht an Licht, deren einfachstes bereits aus einer Storungsrechnung bekannt ist. Fur grose Felder sind die hier abgeleiteten Feldgleichungen von den Maxwell schen sehr verschieden. Sie werden mit den von Born vorgeschlagenen verglichen.
1,982 citations
01 Jan 1934
TL;DR: In this paper, the relation of matter and the electromagnetic field can be interpreted from two opposite standpoints: the unitarian viewpoint assumes only one physical entity (e.g., the EH), and the second viewpoint assumes that the electromagnetic mass is a derived notion expressed by field energy (electromagnetic mass).
Abstract: The relation of matter and the electromagnetic field can be interpreted from two opposite standpoints:— The first which may be called the unitarian standpoint assumes only one physical entity, the electromagnetic field. The particles of matter are considered as singularities of the field and mass is a derived notion to be expressed by field energy (electromagnetic mass).
289 citations
"Nonlinear duality-invariant conform..." refers background in this paper
...[1] M....
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...The Born-Infeld (BI) equations [1], which preserve the electromagnetic duality invariance of Maxwell’s equations, is perhaps the best known example, for reasons reviewed in [2]....
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TL;DR: In this article, the Lagrangian of Born and Infeld was applied to nonlinear electrodynamics and the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts.
Abstract: After a brief discussion of well‐known classical fields we formulate two principles: When the field equations are hyperbolic, particles move along rays like disturbances of the field; the waves associated with stable particles are exceptional. This means that these waves will not transform into shock waves. Both principles are applied to nonlinear electrodynamics. The starting point of the theory is a Lagrangian which is an arbitrary nonlinear function of the two electromagnetic invariants. We obtain the laws of propagation of photons and of charged particles, along with an anisotropic propagation of the wavefronts. The general ``exceptional'' Lagrangian is found. It reduces to the Lagrangian of Born and Infeld when some constant (probably simply connected with the Planck constant) vanishes. A nonsymmetric tensor is introduced in analogy to the Born‐Infeld theory, and finally, electromagnetic waves are compared with those of Einstein‐Schrodinger theory.
275 citations